Summary (11/16/2019) In this project, we develop novel computational methods for image analysis with applications in brain structure variability studies. The proposed methods are based on a new Variational Principle which constructs a deformation with prescribed Jacobian determinant (which models local tissue size changes) and prescribed curl vector (which models local rotations). The goal of this research is to convince the medical image researchers and users that Jacobian determinant as well as curl vector should both be used in all steps of image analysis. Specifically, we develop: (1) A method of averaging a set of deformations based on Jacobian determinants and the curl vectors; the new method constructs the average as a deformation whose Jacobian determinant is equal to the average of the Jacobian determinants and whose curl vector is the average of curl vectors. This new method is biologically meaningful; it also preserves invertibilty of the deformations in the set. (2) A general robust method for construction of unbiased templates from a set of images. The method begins with registering a randomly chosen image in the set to all images in the set. Then the resample of the initial template on the average of the registration deformations is a good approximation; but it may still be biased toward the initial template. We then repeat the averaging process to remove bias and obtain unbiased template. Computational examples are presented to show the effects of curl vector and the effectiveness of method for averaging deformations and our method for construction of unbiased template. The project will significantly enhance our ability to analyze brain image data; improve diagnosis, monitor , and treatment of brain diseases and mental disorder. The project has an important training and educational component. Specifically, a PhD student will participate in algorithm design, computer code development, testing, and software management.